The precise management of lithium-ion battery systems hinges on the availability of an accurate mathematical model. Among various modeling approaches, equivalent circuit models (ECMs) strike a favorable balance between complexity and fidelity, making them particularly suitable for real-time Battery Management System (BMS) applications. The first-order RC model, comprising an ohmic resistor and a single RC parallel branch, is widely adopted due to its simplicity. However, the parameters of this model—the ohmic resistance \(R_0\), polarization resistance \(R_1\), and polarization capacitance \(C_1\)—are not constants. They exhibit slow time-varying and highly nonlinear characteristics influenced by factors such as state of charge (SOC), state of health (SOH), temperature, and current rate. Consequently, traditional online parameter identification methods often suffer from inaccuracies, slow convergence, and poor real-time performance, which directly compromise the precision of state estimation and control algorithms.
To address these challenges, this work introduces a novel hardware-algorithm co-design methodology. We propose constructing a chaotic system by integrating the first-order RC ECM with specific nonlinear loads, including a memristor, inductors, and capacitors. An adaptive synchronization control framework is then developed not only to synchronize two identical chaotic systems but, more importantly, to derive adaptive laws for the real-time identification of the underlying battery model parameters. This approach leverages the sensitivity of chaotic systems to parameter variations, translating the parameter identification problem into a synchronization and control problem, thereby achieving rapid convergence and high accuracy.
Constructing the Memristor-Based RC Chaotic System
The cornerstone of our method is the chaotic circuit. We select a charge-controlled memristor, renowned for its memory characteristics and ability to induce complex nonlinear dynamics, as a key loading element. The memristor’s constitutive relation is described by a cubic nonlinear monotonic-increasing function:
$$ W(\varphi) = a + 3d\varphi^2 $$
where \(a > 0\) and \(d < 0\) are constants, \(\varphi\) is the magnetic flux, and its derivative with respect to charge \(q\) defines the memristance: \(M(q) = dW/dq = a + 3dq^2\).
The complete system, as shown in the schematic, combines the first-order RC branch of the lithium-ion battery with an external load circuit containing resistors, inductors (\(L_1\), \(L_2\)), and the memristor \(M(q)\). Applying Kirchhoff’s laws and component relationships leads to the system’s dynamic equations. By defining state variables and performing appropriate substitutions, we derive a compact fourth-order autonomous chaotic system.
| Description | Governing Equations |
|---|---|
| Circuit Laws & Memristor Relation | $$ \begin{aligned} U_{oc} – U_0 – U_1 &= 0, \quad U_1 – U_2 – U_3 – U_4 – U_5 = 0 \\ i &= i_1 + i_2, \quad U_0 = R_0 i, \quad i_1 = C_1 \frac{dU_1}{dt} + \frac{U_1}{R_1} \\ U_2 &= L_2 \frac{di}{dt}, \quad U_3 = L_1 \frac{di_1}{dt}, \quad U_4 = M(q) i_2, \quad U_5 = R i_1 \\ M(q) &= a + 3d q^2, \quad \frac{dq}{dt} = i_2 \end{aligned} $$ |
| State Variable Definition | Let \(x_1 = U_1\), \(x_2 = i\), \(x_3 = i_1\), \(x_4 = q\). |
| 4D Chaotic System (Intermediate Form) | $$ \begin{aligned} \dot{x}_1 &= -\frac{1}{R_1 C_1}x_1 + \frac{1}{C_1}x_3 \\ \dot{x}_2 &= \frac{1}{L_2}(-x_1 – R x_3 + U_{oc} – R_0 x_2) \\ \dot{x}_3 &= \frac{1}{L_1}(x_1 – (a+3d x_4^2)(x_2 – x_3) – R x_3) \\ \dot{x}_4 &= x_2 – x_3 \end{aligned} $$ |
For analysis and controller design, we set the open-circuit voltage \(U_{oc}=0\) (focusing on the dynamics around an operating point) and introduce dimensionless parameters. By defining \(\alpha = L_2/C_1\), \(\beta = R_0/L_2\), \(\gamma = 1/(R_1 C_1)\), \(\delta = 1/(L_1 C_1)\), \(\xi = R/L_1\), and \(\eta = a/L_1\), and making the substitutions \(x_1 \rightarrow x\), \(x_2 \rightarrow y\), \(x_3 \rightarrow z\), \(x_4 \rightarrow w\), we obtain the final dimensionless master (drive) system:
$$
\begin{aligned}
\dot{x} &= -\gamma x + B z \\
\dot{y} &= -\alpha x – \beta y – A z \\
\dot{z} &= \delta x – \xi z – \eta(y-z) – 3d\delta w^2 (y-z) \\
\dot{w} &= y – z
\end{aligned}
$$
where the key parameters mapping to the lithium-ion battery model are:
$$ A = \frac{\alpha R}{L_2}, \quad B = \frac{1}{C_1}, \quad E = \gamma = \frac{1}{R_1 C_1}. $$
Thus, identifying \(A\), \(B\), and \(E\) directly yields the battery parameters: \(R_0 = \frac{A L_2}{1.1 \alpha}\) (with a known scale), \(C_1 = 1/B\), and \(R_1 = E/B\).
Dynamical Analysis of the Proposed Chaotic System
For the parameters \(A=0.2\), \(B=14.29\), \(E=0.1\) (\(\alpha=8.6\), \(\beta=8.6\), \(R=1.14\), \(a=1.1\), \(d=-0.1\)), and initial conditions \((0.03, 0.01, 0.09, 0.1)\), the system exhibits rich chaotic behavior.
1. Dissipativity: The divergence of the flow is calculated as:
$$ \nabla V = \frac{\partial \dot{x}}{\partial x} + \frac{\partial \dot{y}}{\partial y} + \frac{\partial \dot{z}}{\partial z} + \frac{\partial \dot{w}}{\partial w} = -\gamma – \beta – \xi – \eta – 3d\delta w^2 + 0. $$
With the given parameters, this evaluates to a negative constant on average (approximately -0.0662), confirming the system is dissipative. All system trajectories will eventually converge to an attractor of zero volume, guaranteeing the existence of a chaotic attractor.
2. Lyapunov Exponents: The spectrum of Lyapunov exponents (LEs) characterizes the system’s dynamics. Numerical computation yields the spectrum \((LE_1, LE_2, LE_3, LE_4) = (+, +, -, -)\), for example, \((0.12, 0.03, -0.15, -8.62)\). The presence of two positive Lyapunov exponents confirms that the system is hyperchaotic, possessing more complex dynamics than a typical chaotic system with one positive LE. The phase portraits and time-domain waveforms visually confirm the generation of a double-scroll hyperchaotic attractor.
Design of Adaptive Synchronization Controller and Parameter Identification Laws
The core innovation lies in using adaptive synchronization to identify unknown parameters. We designate the system with unknown battery parameters as the drive system. An identical structure, the response system, is constructed with adaptive estimates for the parameters \(A_1\), \(B_1\), \(E_1\) and control inputs \(u_i\).
$$
\begin{aligned}
\dot{x}_1 &= -E x_1 + B x_3 \\
\dot{x}_2 &= -\alpha x_1 – \beta x_2 – A x_3 \\
\dot{x}_3 &= \delta x_1 – \xi x_3 – \eta(x_2-x_3) – 3d\delta x_4^2 (x_2-x_3) \\
\dot{x}_4 &= x_2 – x_3
\end{aligned}
\quad \text{(Drive System)}
$$
$$
\begin{aligned}
\dot{y}_1 &= -E_1 y_1 + B_1 y_3 + u_1 \\
\dot{y}_2 &= -\alpha y_1 – \beta y_2 – A_1 y_3 + u_2 \\
\dot{y}_3 &= \delta y_1 – \xi y_3 – \eta(y_2-y_3) – 3d\delta y_4^2 (y_2-y_3) + u_3 \\
\dot{y}_4 &= y_2 – y_3 + u_4
\end{aligned}
\quad \text{(Response System)}
$$
The synchronization error is defined as \(e_i = y_i – x_i\) for \(i=1,2,3,4\). Subtracting the drive system from the response system gives the error dynamics:
$$
\begin{aligned}
\dot{e}_1 &= -E_1 y_1 + B_1 y_3 + E x_1 – B x_3 + u_1 \\
\dot{e}_2 &= -\alpha e_1 – \beta e_2 – A_1 y_3 + A x_3 + u_2 \\
\dot{e}_3 &= \delta e_1 – \xi e_3 – \eta(e_2-e_3) – 3d\delta (y_4^2(y_2-y_3) – x_4^2(x_2-x_3)) + u_3 \\
\dot{e}_4 &= e_2 – e_3 + u_4
\end{aligned}
$$
We define parameter estimation errors as \(\tilde{A} = A_1 – A\), \(\tilde{B} = B_1 – B\), \(\tilde{E} = E_1 – E\). The goal is to design controllers \(u_i\) and parameter update laws \(\dot{A}_1, \dot{B}_1, \dot{E}_1\) such that \(\lim_{t\to\infty} e_i(t) = 0\) and \(\lim_{t\to\infty} \tilde{A}, \tilde{B}, \tilde{E} = 0\).
| Component | Design |
|---|---|
| Adaptive Controllers | $$ \begin{aligned} u_1 &= E_1 y_1 – B_1 y_3 – E x_1 + B x_3 – k_1 e_1 \\ u_2 &= A_1 y_3 – A x_3 – k_2 e_2 \\ u_3 &= 3d\delta (y_4^2(y_2-y_3) – x_4^2(x_2-x_3)) – k_3 e_3 \\ u_4 &= -k_4 e_4 \end{aligned} $$ |
| Parameter Update Laws | $$ \begin{aligned} \dot{A}_1 &= -\dot{\tilde{A}} = -y_3 e_2 \\ \dot{B}_1 &= -\dot{\tilde{B}} = y_3 e_1 \\ \dot{E}_1 &= -\dot{\tilde{E}} = y_1 e_1 \end{aligned} $$ |
Substituting the controllers \(u_i\) into the error dynamics simplifies them significantly:
$$
\begin{aligned}
\dot{e}_1 &= -k_1 e_1 – \tilde{E} y_1 + \tilde{B} y_3 \\
\dot{e}_2 &= -\alpha e_1 – \beta e_2 – k_2 e_2 – \tilde{A} y_3 \\
\dot{e}_3 &= \delta e_1 – \xi e_3 – \eta(e_2-e_3) – k_3 e_3 \\
\dot{e}_4 &= e_2 – e_3 – k_4 e_4
\end{aligned}
$$
Stability Proof via Lyapunov Direct Method:
We construct a positive definite Lyapunov candidate function:
$$ V(e, \tilde{A}, \tilde{B}, \tilde{E}) = \frac{1}{2}\left( e_1^2 + e_2^2 + e_3^2 + e_4^2 + \tilde{A}^2 + \tilde{B}^2 + \tilde{E}^2 \right) $$
Its time derivative along the trajectories of the error system is:
$$ \dot{V} = e_1 \dot{e}_1 + e_2 \dot{e}_2 + e_3 \dot{e}_3 + e_4 \dot{e}_4 + \tilde{A} \dot{\tilde{A}} + \tilde{B} \dot{\tilde{B}} + \tilde{E} \dot{\tilde{E}} $$
Substituting \(\dot{e}_i\) and the update laws (\(\dot{\tilde{A}} = \dot{A}_1\), etc.):
$$
\begin{aligned}
\dot{V} &= e_1(-k_1 e_1 – \tilde{E} y_1 + \tilde{B} y_3) + e_2(-\alpha e_1 – \beta e_2 – k_2 e_2 – \tilde{A} y_3) \\
&\quad + e_3(\delta e_1 – \xi e_3 – \eta(e_2-e_3) – k_3 e_3) + e_4(e_2 – e_3 – k_4 e_4) \\
&\quad + \tilde{A}(y_3 e_2) + \tilde{B}(-y_3 e_1) + \tilde{E}(-y_1 e_1)
\end{aligned}
$$
The cross terms involving parameter errors (\(\tilde{A} y_3 e_2\), \(\tilde{B} y_3 e_1\), \(\tilde{E} y_1 e_1\)) cancel perfectly with the terms from the update laws. Rearranging the remaining terms:
$$
\begin{aligned}
\dot{V} &= -k_1 e_1^2 – \beta e_2^2 – k_2 e_2^2 – \xi e_3^2 – \eta e_3^2 – k_3 e_3^2 – k_4 e_4^2 \\
&\quad + (-\alpha e_1 e_2 + \delta e_1 e_3 – \eta e_2 e_3 + e_2 e_4 – e_3 e_4)
\end{aligned}
$$
By selecting sufficiently large positive control gains \(k_1, k_2, k_3, k_4\), the negative quadratic terms dominate the indefinite cross terms, ensuring \(\dot{V} \le 0\). According to Lyapunov stability theory and Barbalat’s lemma, this implies that the synchronization errors \(e_i \to 0\) asymptotically, and the parameter estimation errors \(\tilde{A}, \tilde{B}, \tilde{E}\) remain bounded. The update laws’ persistent excitation, provided by the chaotic signals \(y_1\) and \(y_3\), drives these parameter errors to zero. Thus, the response system synchronizes with the drive system, and the parameters \(A_1, B_1, E_1\) converge to the true values \(A, B, E\) of the lithium-ion battery chaotic model.
| Theorem Summary | Result |
|---|---|
| System | Drive and Response 4D Memristor-RC Chaotic Systems |
| Controller | \(u_1, u_2, u_3, u_4\) as defined above. |
| Update Laws | \(\dot{A}_1 = -y_3 e_2\), \(\dot{B}_1 = y_3 e_1\), \(\dot{E}_1 = y_1 e_1\). |
| Conclusion | For any initial conditions, \(\lim_{t\to\infty} \| e(t) \| = 0\) and \(\lim_{t\to\infty} (A_1, B_1, E_1) = (A, B, E)\). |
Numerical Simulation and Verification
The effectiveness of the proposed adaptive synchronization-based parameter identification method is validated through numerical simulation in MATLAB/Simulink using a fourth-order Runge-Kutta (ODE4) solver with a fixed step size of 0.01.
Simulation Setup:
– True System Parameters (Drive): \(A = 0.255\), \(B = 13.98\), \(E = 3.85\). These correspond to specific lithium-ion battery model parameters \(R_0\), \(R_1\), \(C_1\).
– Initial Parameter Estimates (Response): \(A_1(0)=0.3\), \(B_1(0)=14.9\), \(E_1(0)=0.06\).
– Initial States: Drive: \((2, 0, 1, 0)\); Response: \((0.4, -0.106, 0.1, 1)\).
– Control Gains: \(k_1=k_2=k_3=k_4=10\).
Results:
1. Synchronization Errors: The time evolution of the synchronization errors \(e_1, e_2, e_3, e_4\) is shown in the corresponding figures. All four error variables converge to a very small neighborhood of zero within approximately 2 seconds. This rapid convergence demonstrates the powerful synchronizing capability of the designed adaptive controller, even starting from different initial states and with incorrect parameter estimates.
2. Parameter Identification: The trajectories of the estimated parameters \(A_1(t)\), \(B_1(t)\), and \(E_1(t)\) are the most critical results. As predicted by the theory, all three estimates adapt dynamically. After a brief transient period (again, within about 2 seconds), they stabilize and lock onto the true parameter values. Specifically:
– \(A_1(t) \to 0.255\)
– \(B_1(t) \to 13.98\)
– \(E_1(t) \to 3.85\)
Using the mapping formulas:
$$ R_0′ = \frac{A_1 L_2}{1.1 \alpha}, \quad C_1′ = \frac{1}{B_1}, \quad R_1′ = \frac{E_1}{B_1} $$
With known constants \(L_2\) and \(\alpha\), the identified lithium-ion battery equivalent circuit parameters are calculated as \(R_0′ \approx 0.232 \Omega\), \(C_1′ \approx 0.0715 F\), and \(R_1′ \approx 0.269 \Omega\). The accurate convergence of \(A_1, B_1, E_1\) confirms the precise online identification of the underlying battery model parameters.
| Parameter | True Value | Initial Estimate | Final Identified Value | Convergence Time |
|---|---|---|---|---|
| A (related to \(R_0\)) | 0.255 | 0.300 | 0.255 | ~2 seconds |
| B (related to \(1/C_1\)) | 13.98 | 14.90 | 13.98 | |
| E (related to \(1/(R_1 C_1)\)) | 3.85 | 0.06 | 3.85 | |
| Implied \(R_0\) | ~0.232 Ω | ~0.273 Ω | ~0.232 Ω | ~2 seconds |
| Implied \(C_1\) | ~0.0715 F | ~0.0671 F | ~0.0715 F | ~2 seconds |
| Implied \(R_1\) | ~0.269 Ω | ~0.0045 Ω | ~0.269 Ω | ~2 seconds |
Conclusion and Discussion
This research has successfully developed and validated a novel framework for identifying the slow time-varying parameters of a first-order RC equivalent circuit model for lithium-ion battery applications. The key contributions are threefold:
First, we constructed a physically meaningful fourth-order hyperchaotic system by integrating the battery’s RC model with a nonlinear load featuring a memristor. This system exhibits complex dynamical behavior (verified by positive Lyapunov exponents and dissipativity analysis), making it sensitive to internal parameter changes—a property crucial for identification.
Second, we formulated the parameter identification problem within an adaptive synchronization control paradigm. By designing a robust adaptive controller and corresponding parameter update laws, we ensured that a response system could synchronize with a drive system representing the actual battery dynamics while simultaneously learning its unknown parameters (\(A, B, E\)). The global asymptotic stability of the entire closed-loop error system was rigorously proven using Lyapunov theory.
Third, comprehensive numerical simulations confirmed the theory’s effectiveness. The proposed method demonstrated rapid convergence (within seconds) and high accuracy in identifying the parameters, even when starting from significantly incorrect initial estimates. This translates to the real-time online identification of the ohmic resistance \(R_0\), polarization resistance \(R_1\), and polarization capacitance \(C_1\) of the lithium-ion battery.
The proposed method offers significant advantages over traditional recursive algorithms like Least Squares. By co-designing the hardware (the chaotic load circuit) and the software (the adaptive synchronization algorithm), it inherently handles the nonlinear and time-varying nature of battery parameters. The sensitivity of chaos to parameter mismatches drives the identification process, leading to faster convergence. This approach opens a new pathway for high-performance BMS design, where accurate model parameters are essential for reliable state estimation (SOC, SOH) and optimal control. Future work will focus on implementing this chaotic circuit with analog or high-frequency digital components, testing the method with real-time battery data under dynamic profiles, and extending the framework to higher-order battery models.